Slenderness Ratio: Difference between revisions

Latest revision as of 10:47, 7 September 2026

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Slenderness ratio

Fundamentals

The slenderness ratio is used to evaluate the geometric suitability of test specimens for determining material values in compression tests and for calculating the critical buckling stress.

When subjected to compressive stress, the test specimen must not buckle, otherwise an EULERian stability failure will occur. To ensure this with sufficient slenderness, a slenderness ratio of λ = 10 is preferable, but at least λ = 6 must be maintained.

The slenderness ratio λ is the ratio of the test specimen height h to the smallest radius of inertia of its base area:

\lambda \,=\,{\frac {h}{i}}

with

i\,=\,{\sqrt {\frac {I_{y}}{A_{0}}}}

prism:

I_{y}\,=\,{\frac {bd^{3}}{12}}

A_{0}\,=\,b\cdot d

\Rightarrow

h\,=\,{\frac {d}{3{,}46}}\cdot \lambda

cylinder:

I_{p}\,=\,{\frac {\pi d^{4}}{64}}

A_{0}\,=\,{\frac {\pi }{4}}\cdot d^{2}

\Rightarrow

h\,=\,{\frac {d}{4}}\cdot \lambda

with

i		smallest radius of inertia
I_y		smallest axial moment of inertia
I_p		polar moment of inertia
A₀		specimen base area
h		specimen height
b		specimen width
d		thickness or diameter

References

Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 125–133 (ISBN 978-1-56990-548-7; ePub: ISBN 978-1-56990-808-2; E-Book: E-book: 978-1-56990-807-5; see AMK-Library under A 22)

@@ Line 15: / Line 15: @@
 |-
 |width="20px"|
-|width="500px" | <math>\lambda\,=\,\frac{h}{i}</math> &nbsp;&nbsp;mit&nbsp;&nbsp; <math>i\,=\,\sqrt{\frac{I_y}{A}}</math>
+|width="500px" | <math>\lambda\,=\,\frac{h}{i}</math> &nbsp;&nbsp;with&nbsp;&nbsp; <math>i\,=\,\sqrt{\frac{I_y}{A_0}}</math>
 |}
@@ Line 22: / Line 22: @@
 |-
 |width="20px"|
-|width="500px" | <math>I_y\,=\,\frac{bd^3}{12}</math> <math>A_0\,=\,b\cdot d</math> <math>\Rightarrow</math> <math>h\,=\,\frac{d}{3{,}46}\cdot\lambda</math>
+|width="500px" | <math>I_y\,=\,\frac{bd^3}{12}</math> &nbsp;&nbsp;&nbsp;&nbsp; <math>A_0\,=\,b\cdot d</math> &nbsp;&nbsp;&nbsp;&nbsp; <math>\Rightarrow</math> &nbsp;&nbsp;&nbsp;&nbsp; <math>h\,=\,\frac{d}{3{,}46}\cdot\lambda</math>
 |}
@@ Line 29: / Line 29: @@
 |-
 |width="20px"|
-|width="500px" |<math>I_p\,=\,\frac{\pi d^4}{64}</math> <math>A_0\,=\,\frac{\pi}{4}\cdot d^2</math> <math>\Rightarrow</math> <math>h\,=\,\frac{d}{4}\cdot \lambda</math>
+|width="500px" |<math>I_p\,=\,\frac{\pi d^4}{64}</math> &nbsp;&nbsp;&nbsp;&nbsp; <math>A_0\,=\,\frac{\pi}{4}\cdot d^2</math> <math>\Rightarrow</math> &nbsp;&nbsp;&nbsp;&nbsp; <math>h\,=\,\frac{d}{4}\cdot \lambda</math>
 |}
@@ Line 72: / Line 72: @@
 * [[Creep Behaviour – Creep Compression Test]]
+==References==
-'''References'''
+* [[Grellmann, Wolfgang|Grellmann, W.]], [[Seidler, Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 125–133 (ISBN 978-1-56990-548-7; ePub: ISBN 978-1-56990-808-2; E-Book: E-book: 978-1-56990-807-5; see [[AMK-Library]] under A 22)
-* [[Grellmann, Wolfgang|Grellmann, W.]], [[Seidler, Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 125–133 (ISBN 978-1-56990-548-7; ePub: ISBN 978-1-56990-808-2; E-Book: E-book: 978-1-56990-807-5; see [[AMK-Büchersammlung|AMK-Library]] under A 22)
 [[category:Compression Test]]

Latest revision as of 10:47, 7 September 2026

Fundamentals

See also

References